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Sum of series/ (xln(x)sin(x))/(x^6-1)

Sum of series (xln(x)sin(x))/(x^6-1)


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The solution

You have entered [src] 
  oo                 
____                 
\   `                
 \    x*log(x)*sin(x)
  \   ---------------
  /         6        
 /         x  - 1    
/___,                
n = 1
$$\sum_{n=1}^{\infty} \frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}$$ 
Sum(((x*log(x))*sin(x))/(x^6 - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} 1$$
Let's take the limit
we find
True

False
The answer [src] 
oo*x*log(x)*sin(x)
------------------
           6      
     -1 + x
$$\frac{\infty x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}$$ 
oo*x*log(x)*sin(x)/(-1 + x^6)

    Examples of finding the sum of a series

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